$R$-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

This paper studies R-equivalence on cubic surfaces (3D algebraic varieties defined by cubic equations) over p-adic fields, proving that for certain special types of surfaces with good reduction properties, R-equivalence is either trivial or has limited structure (exponent 2). The authors resolve long-standing questions about specific cubic surfaces, including one posed by Manin over 50 years ago, by developing new mathematical methods and notably incorporating AI assistance (AlphaEvolve and Gemini 3 Deep Think) to help prove lemmas and explore the problem space.